In the video game, if an adversary plays a spy against you and it is cleaned at the end of the round, it's not in your discard pile in the next round. So you can't for example revive it using the medic function.

But I if it went to the adversary's discard pile, they could it revive it themselves the next round. I can't remember, how that worked in the video game. Do you? Or is there some rule in the booklet I overread?

Maybe it would be best to keep played spies in seperate discard piles after cleaning the board. Their cover's blown anyway, isn't it?

I'm pretty sure it plays the same way it does in the VG. When you play a spy, you play it on the opponents field. They then have control of the card. If it is destroyed or at the end of a round it goes into it's CURRENT owner's graveyard. From there it can be brought back just like any other card that can be affected by other cards' abilities.